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    <title>dsaupd</title>
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    <center>Scilab Function</center>
    <div align="right">Last update : 25/10/2006</div>
    <p>
      <b>dsaupd</b> - 
		Interface for the Implicitly Restarted Arnoldi Iteration,  to compute approximations to a 
		few eigenpairs of a real and symmetric linear operator
	</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[IDO,RESID,V,IPARAM,IPNTR,WORKD,WORKL,INFO] = dsaupd(ID0,BMAT,N,WHICH,NEV,TOL,RESID,NCV,V,IPARAM,IPNTR,WORKD,WORKL,INFO)</tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>ID0</b>
        </tt>
        <p>
						: Integer.  (INPUT/OUTPUT) Reverse communication flag.  IDO must be zero on the first 
						call to dsaupd .  IDO will be set internally to
						indicate the type of operation to be performed.  Control is
						then given back to the calling routine which has the
						responsibility to carry out the requested operation and call
						dsaupd  with the result.  The operand is given in
						WORKD(IPNTR(1)), the result must be put in WORKD(IPNTR(2)).
						(If Mode = 2 see remark 5 below)
					</p>
        <p> 
						IDO =  0: first call to the reverse communication interface
					</p>
        <p> 
						IDO = -1: compute  Y = OP * X  where
						IPNTR(1) is the pointer into WORKD for X,
						IPNTR(2) is the pointer into WORKD for Y.
						This is for the initialization phase to force the
						starting vector into the range of OP.
					</p>
        <p> 
						IDO =  1: compute  Y = OP * X where
						IPNTR(1) is the pointer into WORKD for X,
						IPNTR(2) is the pointer into WORKD for Y.
						In mode 3,4 and 5, the vector B * X is already
						available in WORKD(ipntr(3)).  It does not
						need to be recomputed in forming OP * X.
					</p>
        <p> 
						IDO =  2: compute  Y = B * X  where
						IPNTR(1) is the pointer into WORKD for X,
						IPNTR(2) is the pointer into WORKD for Y.
					</p>
        <p> 
						IDO =  3: compute the IPARAM(8) shifts where
						IPNTR(11) is the pointer into WORKL for
						placing the shifts. See remark 6 below.
					</p>
        <p> 
						IDO = 99: done
					</p>
      </li>
      <li>
        <tt>
          <b>BMAT</b>
        </tt>
						: Character, specifies the type of the matrix B that defines the
						semi-inner product for the operator OP.
					<p> 
						B = 'I' -&gt; standard eigenvalue problem A*x = lambda*x
					</p>
        <p> 
						B = 'G' -&gt; generalized eigenvalue problem A*x = lambda*B*x
					</p>
      </li>
      <li>
        <tt>
          <b>N</b>
        </tt>
						: Integer, dimension of the eigenproblem.
					</li>
      <li>
        <tt>
          <b>WHICH</b>
        </tt>
						: string of length 2, Specify which of the Ritz values of OP to compute.
					<p>
						'LA' - compute the NEV largest (algebraic) eigenvalues.
					</p>
        <p> 
						'SA' - compute the NEV smallest (algebraic) eigenvalues.
					</p>
        <p> 
						'LM' - compute the NEV largest (in magnitude) eigenvalues.
					</p>
        <p> 
						'SM' - compute the NEV smallest (in magnitude) eigenvalues. 
					</p>
        <p> 
						'BE' - compute NEV eigenvalues, half from each end of the
						spectrum.  When NEV is odd, compute one more from the
						high end than from the low end.
						(see remark 1 below)
					</p>
      </li>
      <li>
        <tt>
          <b>NEV</b>
        </tt>
						: Integer, number of eigenvalues of OP to be computed. 0 &lt; NEV  &lt; N.
					</li>
      <li>
        <tt>
          <b>TOL</b>
        </tt>
        <p>
						4. B.N. Parlett, B. Nour-Omid, "Towards a Black Box Lanczos Program",
						Computer Physics Communications, 53 (1989), pp 169-179.
					</p>
        <p>
						5. B. Nour-Omid, B.N. Parlett, T. Ericson, P.S. Jensen, "How to
						Implement the Spectral Transformation", Math. Comp., 48 (1987),
						pp 663-673.
					</p>
        <p>
						6. R.G. Grimes, J.G. Lewis and H.D. Simon, "A Shifted Block Lanczos 
						Algorithm for Solving Sparse Symmetric Generalized Eigenproblems", 
						SIAM J. Matr. Anal. Apps.,  January (1993).
					</p>
        <p>
						7. L. Reichel, W.B. Gragg, "Algorithm 686: FORTRAN Subroutines
						for Updating the QR decomposition", ACM TOMS, December 1990,
						Volume 16 Number 4, pp 369-377.
					</p>
        <p>
						8. R.B. Lehoucq, D.C. Sorensen, "Implementation of Some Spectral
						Transformations in a k-Step Arnoldi Method". In Preparation.
					</p>
						: scalar.   Stopping criterion: the relative accuracy of the Ritz value 
						is considered acceptable if BOUNDS(I) &lt;= TOL*ABS(RITZ(I)).
						If TOL &lt;= 0. is passed the machine precision is set.
					</li>
      <li>
        <tt>
          <b>RESID</b>
        </tt>
					: array of length N (INPUT/OUTPUT)
					<p>
						On INPUT: 
						If INFO==0, a random initial residual vector is used, else RESID contains the initial residual vector,
							possibly from a previous run.
					</p>
        <p>
						On OUTPUT:
						RESID contains the final residual vector. 
					</p>
      </li>
      <li>
        <tt>
          <b>NCV</b>
        </tt>
					: Integer.Number of columns of the matrix V (less than or equal to N).
						This will indicate how many Lanczos vectors are generated 
						at each iteration.  After the startup phase in which NEV 
						Lanczos vectors are generated, the algorithm generates 
						NCV-NEV Lanczos vectors at each subsequent update iteration.
						Most of the cost in generating each Lanczos vector is in the 
						matrix-vector product OP*x. (See remark 4 below).
					</li>
      <li>
        <tt>
          <b>V</b>
        </tt>
						: N by NCV array. The NCV columns of V contain the Lanczos basis vectors.
					</li>
      <li>
        <tt>
          <b>IPARAM</b>
        </tt>
					: array of length 11.  (INPUT/OUTPUT)
				<p>
					IPARAM(1) = ISHIFT: method for selecting the implicit shifts.
					The shifts selected at each iteration are used to restart
					the Arnoldi iteration in an implicit fashion.
				</p>
        <p>
					ISHIFT = 0: the shifts are provided by the user via
								reverse communication.  The NCV eigenvalues of
								the current tridiagonal matrix T are returned in
								the part of WORKL array corresponding to RITZ.
								See remark 6 below.
				</p>
        <p>
					ISHIFT = 1: exact shifts with respect to the reduced 
								tridiagonal matrix T.  This is equivalent to 
								restarting the iteration with a starting vector 
								that is a linear combination of Ritz vectors 
								associated with the "wanted" Ritz values.
				</p>
        <p>
					IPARAM(2) = LEVEC
					No longer referenced. See remark 2 below.
				</p>
        <p>
					IPARAM(3) = MXITER
					On INPUT:  maximum number of Arnoldi update iterations allowed. 
					On OUTPUT: actual number of Arnoldi update iterations taken. 
				</p>
        <p>
					IPARAM(4) = NB: blocksize to be used in the recurrence.
					The code currently works only for NB = 1.
				</p>
        <p>
					IPARAM(5) = NCONV: number of "converged" Ritz values.
					This represents the number of Ritz values that satisfy
					the convergence criterion.
				</p>
        <p>
					IPARAM(6) = IUPD
					No longer referenced. Implicit restarting is ALWAYS used. 
				</p>
        <p>
					IPARAM(7) = MODE
					On INPUT determines what type of eigenproblem is being solved.
					Must be 1,2,3,4,5; See under Description of dsaupd  for the 
					five modes available.
				</p>
        <p>
					IPARAM(8) = NP
					When ido = 3 and the user provides shifts through reverse
					communication (IPARAM(1)=0), dsaupd  returns NP, the number
					of shifts the user is to provide. 0  &lt; NP  &lt;= NCV-NEV. See Remark
					6 below.
				</p>
        <p>
					IPARAM(9) = NUMOP, IPARAM(10) = NUMOPB, IPARAM(11) = NUMREO,
					OUTPUT: NUMOP  = total number of OP*x operations,
							NUMOPB = total number of B*x operations if BMAT='G',
							NUMREO = total number of steps of re-orthogonalization.        
			
				</p>
      </li>
      <li>
        <tt>
          <b>IPNTR</b>
        </tt>
						: array of length 11.  Pointer to mark the starting locations in the WORKD and WORKL
						arrays for matrices/vectors used by the Lanczos iteration.
					<p>
						IPNTR(1): pointer to the current operand vector X in WORKD.
					</p>
        <p>
						IPNTR(2): pointer to the current result vector Y in WORKD.
					</p>
        <p>
						IPNTR(3): pointer to the vector B * X in WORKD when used in 
						the shift-and-invert mode.
					</p>
        <p>
						IPNTR(4): pointer to the next available location in WORKL
						that is untouched by the program.
					</p>
        <p>
						IPNTR(5): pointer to the NCV by 2 tridiagonal matrix T in WORKL.
					</p>
        <p>
						IPNTR(6): pointer to the NCV RITZ values array in WORKL.
					</p>
        <p>
						IPNTR(7): pointer to the Ritz estimates in array WORKL associated
						with the Ritz values located in RITZ in WORKL.
					</p>
        <p>
						IPNTR(11): pointer to the NP shifts in WORKL. See Remark 6 below.
					</p>
        <p>
						Note: IPNTR(8:10) is only referenced by dseupd . See Remark 2.
					</p>
        <p>
						IPNTR(8): pointer to the NCV RITZ values of the original system.
					</p>
        <p>
						IPNTR(9): pointer to the NCV corresponding error bounds.
					</p>
        <p>
						IPNTR(10): pointer to the NCV by NCV matrix of eigenvectors
						of the tridiagonal matrix T. Only referenced by
						dseupd  if RVEC = .TRUE. See Remarks
					</p>
      </li>
      <li>
        <tt>
          <b>WORKD</b>
        </tt>
					: work array of length 3*N.  (REVERSE COMMUNICATION)
						Distributed array to be used in the basic Arnoldi iteration
						for reverse communication.  The user should not use WORKD 
						as temporary workspace during the iteration. Upon termination
						WORKD(1:N) contains B*RESID(1:N). If the Ritz vectors are desired
						subroutine dseupd  uses this output.
						See Data Distribution Note below. 
					</li>
      <li>
        <tt>
          <b>WORKL</b>
        </tt>
						: work array of length at least NCV**2 + 8*NCV.  (OUTPUT/WORKSPACE)
						Private (replicated) array on each PE or array allocated on
						the front end.  See Data Distribution Note below. add here the parameter description
					</li>
      <li>
        <tt>
          <b>INFO</b>
        </tt>
						: Integer.  (INPUT/OUTPUT)
					<p>
						If INFO == 0, a randomly initial residual vector is used, else RESID contains the initial residual vector,
						possibly from a previous run.
					</p>
        <p>
						Error flag on output.
					</p>
        <p>
						=  0: Normal exit.
					</p>
        <p>
					=  1: Maximum number of iterations taken.
						All possible eigenvalues of OP has been found. IPARAM(5)  
						returns the number of wanted converged Ritz values.
					</p>
        <p>
					=  2: No longer an informational error. Deprecated starting
						with release 2 of ARPACK.
					</p>
        <p>
					=  3: No shifts could be applied during a cycle of the 
						Implicitly restarted Arnoldi iteration. One possibility 
						is to increase the size of NCV relative to NEV. 
						See remark 4 below.
					</p>
        <p>
						= -1: N must be positive.
					</p>
        <p>
						= -2: NEV must be positive.
					</p>
        <p>
						= -3: NCV must be greater than NEV and less than or equal to N.
					</p>
        <p>
						= -4: The maximum number of Arnoldi update iterations allowed
						must be greater than zero.
					</p>
        <p>
						= -5: WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.
					</p>
        <p>
						= -6: BMAT must be one of 'I' or 'G'.
					</p>
        <p>
						= -7: Length of private work array WORKL is not sufficient.
					</p>
        <p>
						= -8: Error return from trid. eigenvalue calculation;
							Informatinal error from LAPACK routine dsteqr .
					</p>
        <p>
						= -9: Starting vector is zero.
					</p>
        <p>
						= -10: IPARAM(7) must be 1,2,3,4,5.
					</p>
        <p>
						= -11: IPARAM(7) = 1 and BMAT = 'G' are incompatable.
					</p>
        <p>
						= -12: IPARAM(1) must be equal to 0 or 1.
					</p>
        <p>
						= -13: NEV and WHICH = 'BE' are incompatable.
					</p>
        <p>
						= -9999: Could not build an Arnoldi factorization.
							IPARAM(5) returns the size of the current Arnoldi
							factorization. The user is advised to check that
							enough workspace and array storage has been allocated.
					</p>
      </li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
			Reverse communication interface for the Implicitly Restarted Arnoldi 
			Iteration.  For symmetric problems this reduces to a variant of the Lanczos 
			method.  This method has been designed to compute approximations to a 
			few eigenpairs of a linear operator OP that is real and symmetric 
			with respect to a real positive semi-definite symmetric matrix B, 
			i.e.<tt>
        <b>B*OP = (OP`)*B</b>
      </tt>.
		</p>
    <p>
			Another way to express this condition is
			<tt>
        <b>&lt; x,OPy &gt; = &lt; OPx,y &gt;  where &lt;z,w &gt; = z`Bw </b>
      </tt>.
		</p>
    <p>In the standard eigenproblem B is the identity matrix.( A` denotes transpose of A)</p>
    <p>The computed approximate eigenvalues are called Ritz values and
		the corresponding approximate eigenvectors are called Ritz vectors.</p>
    <p>
			dsaupd  is usually called iteratively to solve one of the 
			following problems:
		</p>
    <dl>
      <dd>
        <b></b>
        <p>
					Mode 1:  A*x = lambda*x, A symmetric 
					===&gt; OP = A  and  B = I.
				</p>
      </dd>
      <dd>
        <b></b>
        <p>
					Mode 2:  A*x = lambda*M*x, A symmetric, M symmetric positive definite
					===&gt; OP = inv[M]*A  and  B = M.
					===&gt; (If M can be factored see remark 3 below)
				</p>
      </dd>
      <dd>
        <b></b>
        <p>
					Mode 3:  K*x = lambda*M*x, K symmetric, M symmetric positive semi-definite
					===&gt; OP = (inv[K - sigma*M])*M  and  B = M. 
					===&gt; Shift-and-Invert mode
				</p>
      </dd>
      <dd>
        <b></b>
        <p>
					Mode 4:  K*x = lambda*KG*x, K symmetric positive semi-definite, 
					KG symmetric indefinite
					===&gt; OP = (inv[K - sigma*KG])*K  and  B = K.
					===&gt; Buckling mode
				</p>
      </dd>
      <dd>
        <b></b>
        <p>
					Mode 5:  A*x = lambda*M*x, A symmetric, M symmetric positive semi-definite
					===&gt; OP = inv[A - sigma*M]*[A + sigma*M]  and  B = M.
					===&gt; Cayley transformed mode
				</p>
      </dd>
    </dl>
    <p>
			NOTE: The action of w &lt;- inv[A - sigma*M]*v or w &lt;- inv[M]*v
			should be accomplished either by a direct method
			using a sparse matrix factorization and solving
			<tt>
        <b>[A - sigma*M]*w = v  or M*w = v</b>
      </tt>,
		</p>
    <p>
			or through an iterative method for solving these
			systems.  If an iterative method is used, the
			convergence test must be more stringent than
			the accuracy requirements for the eigenvalue
			approximations.
		</p>
    <h3>
      <font color="blue">Remarks</font>
    </h3>
    <dl>
      <p>
			1. The converged Ritz values are always returned in ascending 
				algebraic order.  The computed Ritz values are approximate
				eigenvalues of OP.  The selection of WHICH should be made
				with this in mind when Mode = 3,4,5.  After convergence, 
				approximate eigenvalues of the original problem may be obtained 
				with the ARPACK subroutine dseupd . 
		</p>
      <p>
			2. If the Ritz vectors corresponding to the converged Ritz values
				are needed, the user must call dseupd  immediately following completion
				of dsaupd . This is new starting with version 2.1 of ARPACK.
		</p>
      <p>
			3. If M can be factored into a Cholesky factorization M = LL`
				then Mode = 2 should not be selected.  Instead one should use
				Mode = 1 with  OP = inv(L)*A*inv(L`).  Appropriate triangular 
				linear systems should be solved with L and L` rather
				than computing inverses.  After convergence, an approximate
				eigenvector z of the original problem is recovered by solving
				L`z = x  where x is a Ritz vector of OP.
		</p>
      <p>
			4. At present there is no a-priori analysis to guide the selection
				of NCV relative to NEV.  The only formal requrement is that NCV &gt; NEV.
				However, it is recommended that NCV &gt;= 2*NEV.  If many problems of
				the same type are to be solved, one should experiment with increasing
				NCV while keeping NEV fixed for a given test problem.  This will 
				usually decrease the required number of OP*x operations but it
				also increases the work and storage required to maintain the orthogonal
				basis vectors.   The optimal "cross-over" with respect to CPU time
				is problem dependent and must be determined empirically.
		</p>
      <p>
			5. If IPARAM(7) = 2 then in the Reverse commuication interface the user
				must do the following. When IDO = 1, Y = OP * X is to be computed.
				When IPARAM(7) = 2 OP = inv(B)*A. After computing A*X the user
				must overwrite X with A*X. Y is then the solution to the linear set
				of equations B*Y = A*X.
		</p>
      <p>
			6. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the 
				NP = IPARAM(8) shifts in locations: 
				1   WORKL(IPNTR(11))           
				2   WORKL(IPNTR(11)+1)         
				
				NP  WORKL(IPNTR(11)+NP-1). 
				
				The eigenvalues of the current tridiagonal matrix are located in 
				WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1). They are in the
				order defined by WHICH. The associated Ritz estimates are located in
				WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8)+NCV-1).
		</p>
    </dl>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="dnaupd.htm">
        <tt>
          <b>dnaupd</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
    <h3>
      <font color="blue">Authors</font>
    </h3>
    <dl>
      <dd>
        <b> Danny Sorensen, Richard Lehoucq, Phuong Vu </b>
			CRPC / Rice University  Applied Mathematics  Rice University           
			Houston, Texas   
		</dd>
    </dl>
    <h3>
      <font color="blue">Bibliography</font>
    </h3>
			1. D.C. Sorensen, "Implicit Application of Polynomial Filters in
				a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992),
				pp 357-385.
		<p>
			2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly 
				Restarted Arnoldi Iteration", Rice University Technical Report
				TR95-13, Department of Computational and Applied Mathematics.
		</p>
    <p>
			3. B.N. Parlett and Y. Saad, "Complex Shift and Invert Strategies for
				Real Matrices", Linear Algebra and its Applications, vol 88/89,
				pp 575-595, (1987).  
		</p>
    <h3>
      <font color="blue">Used Function</font>
    </h3>Based on ARPACK routine dnaupd</body>
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